jmc@wuphys.UUCP (Jimmy Chen) (12/03/85)
Since there is still so much confusion on the
slingshot effect (including the mistaken idea that fuel
is burned at the bottom of the trajectory), I will
present an analysis one of the simplest slingshot case.
Namely, the case of zero angular momentum of the satellite
about the planet.
Consider an infinitely mass planet as our scattering
center (i.e. so massive that its velocity is negligible).
As the time the satellite spends in the effective
gravitational well of the planet is short, we can replace
the actual trajectory of the planet with a straight line
trajectory. Thus the planet is moving at a constant
velocity, V, in the x-direction relative to the sun. Let
the satellite be approaching from plus infinity. Its velocity
at infinity relative to the planet is then -v0. Its angular
momentum is zero. By conservation of angular momentum,
the outgoing velocity must also be in the x-direction.
By conservation of energy, the outgoing speed of the
satellite is the same as the incoming speed. To see
this, note that the total energy is the sum of the
kinetic & potential. At infinity, the potential energy
is zero. Thus (KE before = KE after) => (v before = v after).
and the outgoing velocity is +v0. The effect of the planet on
the satellite was only to change the direction of the
motion and not the speed.
Now transform this to the frame of reference of
the sun. The incoming speed is -v0+V while the
outgoing speed is v0+V. The satellite's speed has been
increased by 2v0. One of the most interesting case is
when v0=V. Then, from the sun's frame, the satellite is
sitting still, the planet comes rolling along, grabs the
satellite and throw it forward with a speed 2V.
Note that there is no violation of linear momentum
only a transfer of momentum from the planet to the
satellite. We assumed an infinitely massive planet and
so side stepped this.
Jimmy Chendgary@ecsvax.UUCP (D Gary Grady) (12/04/85)
In article <407@wuphys.UUCP> jmc@wuphys.UUCP (Jimmy Chen) writes: > Consider an infinitely mass planet as our scattering >center (i.e. so massive that its velocity is negligible). This is one of the most amusing simplifying assumptions I have ever seen! Since all orbits end at the Schwarschild radius, can we all go home early today? :-) (No offense, Jimmy, I know what you meant to say.) -- D Gary Grady Duke U Comp Center, Durham, NC 27706 (919) 684-3695 USENET: {seismo,decvax,ihnp4,akgua,etc.}!mcnc!ecsvax!dgary